Condorcet Criterion

The Condorcet candidate, Pairwise Champion (PC), or Condorcet winner (CW) of an election is the candidate who, when compared in turn with each of the other candidates, is preferred over the other candidate. On a one-dimensional political spectrum, the pairwise champion will be at the position of the median voter. Mainly because of Condorcet's voting paradox, a pairwise champion will not always exist in a given set of votes. If the pairwise champion exists, they will be the only candidate in the Smith set; otherwise, the Smith set will have three or more members.

The Condorcet criterion for a voting system is that it chooses the pairwise champion when one exists. Any method conforming to the Condorcet criterion is known as a Condorcet method.

It has been recently argued that the definition of the verb "beat" should be regarded as external to the Condorcet Criterion...and that "beat should be defined in a way that interprets equal-top ranking consistent with the actual preferences, intent and wishes of the equal-top-ranking voters. When such a definition of "beat" is used in the Condorcet Criterion definition, then the Condorcet Criterion is compatible with FBC, and there are Condorcet methods that pass FBC. Discussion and arguments on that matter can be found at the Symmetrical ICT article.

Commentary

Non-ranking methods such as plurality and approval cannot comply with the Condorcet criterion because they do not allow each voter to fully specify their preferences. But instant-runoff voting allows each voter to rank the candidates, yet it still does not comply. A simple example will prove that IRV fails to comply with the Condorcet criterion.

Consider, for example, the following vote count of preferences with three candidates {A,B,C}:

499:

A>B>C

498:

C>B>A

3:

B>C>A

In this case, B is preferred to A by 501 votes to 499, and B is
preferred to C by 502 to 498, hence B is preferred to both A and C. So
according to the Condorcet criteria, B should win. By contrast, according to the rules of IRV, B is ranked first by the fewest voters and is eliminated, and C wins with the transferred voted from B; in plurality voting A wins with the most first choices.

Range voting does not comply because it allows for the difference between 'rankings' to matter. E.g. 51 people might rate A at 100, and B at 90, while 49 people rate A at 0, and B at 100. Condorcet would consider this 51 people voting A>B, and 49 voting B>A, and A would win. Range voting would see this as A having support of 5100/100 = 51%, and B support of (51*90+49*100)/100 = 94.9%; range voting advocates would probably say that in this case the Condorcet winner is not the socially ideal winner.